APPROXIMATION BY CERTAIN POSITIVE LINEAR METHODS OF CONVERGENCE

Abstract

The present thesis deals with the approximation properties of some well-known linear positive operators and their new generalizations which include the Stancu type generalization, bivariate generalization, B ezier variant and q􀀀variant of the well known operators. We divide the thesis into eight chapters. In Chapter 0, we mention literature survey, basic de nitions and some notations of approximation techniques which we have used throughout the thesis. In Chapter 1; we de ne a general sequence of linear positive operators and discuss their approximation behaviour e.g. rate of convergence in ordinary and simultaneous approximation and the estimate of the rate of convergence for functions having a derivative equivalent to a function of bounded variation. Further, we illustrate the convergence of these operators and their rst order derivatives by graphics using Matlab algorithms. In Chapter 2; we consider a one parameter family of hybrid operators and study the local, weighted approximation results, simultaneous approximation of derivatives and statistical convergence. Also, we show the rate of convergence of these operators to a certain function by illustrative graphics in Matlab. The third chapter involves the Kantorovich modi cation of generalized Baskakov operators. We obtain some direct results and then study weighted approximation, simultaneous approximation and statistical convergence properties for these operators. We also obtain the rate of convergence for functions having a derivative equivalent with a function of bounded variation for these operators. Further, we de- ne the bivariate extension of the generalized Baskakov Kantorovich operators and discuss the results on the degree of approximation, asymptotic theorem, order of approximation using Peetre's K􀀀functional and simultaneous approximation for rst order derivatives of operators in polynomial weighted spaces. Lastly, we also show i ii the convergence of the bivariate operators to a certain function and demonstrate the comparison with the bivariate Sz asz-Kantorovich operators through graphics using Matlab algorithm. In Chapter 4; we study some approximation properties of the B ezier variant of generalized Baskakov Kantorovich operators. We obtain direct theorem by means of the rst order modulus of smoothness and the rate of convergence for the functions having a derivative of bounded variation. The fth and sixth chapters deal with the q􀀀analogues of general Gamma type operators and the Stancu generalization of Sz asz-Baskakov operators respectively. First, we obtain the moments of the operators by using the q􀀀beta function and then prove the basic convergence theorem. The Voronovskaja type theorem, local and direct results and weighted approximation in terms of modulus of continuity have been discussed for both of these operators. Lastly, we study the King type approach in order to obtain the better approximation for both of these operators. In the last chapter, we introduce the complex case of the Sz asz-Durrmeyer- Chlodowsky operators and obtain the upper estimate, Voronovskaja type result, the exact order in simultaneous approximation and asymptotic result with quantitative estimates. In this way, we show the overconvergence phenomenon for these operators, namely the extensions of approximation properties orders of these convergencies to sets in the complex plane that contain the interval [0;1):